Methods and apparatus for generation and control of coherent polarization mode dispersion

ABSTRACT

Methods and apparatus for coherent PMD generation are provided. A PMD generator can include at least four birefringent stages in optical series, thereby forming at least three pairs of adjacent stages. Each of the stages includes a harmonic differential group delay element and a phase-compensating element. The generator can be made colorless (i.e., made to have the same PMD at each WDM channel) and can be operated such that DGD and second order PMD can be independently generated and controlled. These PMD generators can be used in PMD compensators and PMD emulators.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This claims the benefit of U.S. Provisional Patent ApplicationNo. 60/251,765, filed Dec. 7, 2000, Application No. 60/259,913, filedJan. 5, 2001, and Application No. 60/275,914, filed Mar. 15, 2001, andPatent Application No. which are hereby incorporated by reference hereinin their entireties.

FIELD OF THE INVENTION

[0002] This invention relates to the generation of polarization modedispersion, and more particularly to methods and apparatus forcoherently generating polarization mode dispersion, aligning a coherentpolarization mode dispersion spectrum to a wavelength-divisionmultiplexed (hereinafter, “WDM”) channel grid, and controllinggeneration of first and second order polarization mode dispersion acrossa WDM channel bandwidth.

BACKGROUND OF THE INVENTION

[0003] Polarization mode dispersion (hereinafter, “PMD”) is an opticalproperty that can be generated by a concatenation of two or morebirefringent elements. PMD can be a significant impairment in highdata-rate optical communication systems when the transmission medium isoptical fiber. Data transmission rates that are effected by the PMD ofoptical fiber are typically 10 Gbps, 40 Gbps, and higher.

[0004] Optical fiber can exhibit PMD because of imperfections within thefiber, which induce localized birefringence. When the transmission pathis long, these localized birefringent sections can combine to yield aparticularly complicated polarization-dependent effect. These localizedsections are known to result, for example, from eccentricities of thewaveguide's core, micro-bubbles in the waveguide core and/or cladding,and strain gradients through the fiber cross-section. Mechanical stresson the fiber resulting from cabling and installation can also cause thefiber to suffer stress-induced birefringence. Environmental changesexperienced by a fiber can be dynamic and statistical in nature, and arebelieved to result in PMD changes that can last for variable periods oftime and vary with wavelength, with the potential for prolongeddegradation of data transmission.

[0005] In the laboratory and the field, there are reasons toartificially generate PMD in a controlled fashion.

[0006] In the laboratory, for example, a PMD emulator is desirably usedto predictably and repeatably add PMD to signals generated by opticaltransmitters for testing optical receivers. In many cases, however, thecenter frequency of the optical signal being tested may not be properlyaligned with the PMD spectrum generated by the emulator. Because aconventional PMD emulator cannot controllably “frequency shift” itsspectrum to accommodate for the misalignment, those attempting toevaluate the PMD response of receivers and other equipment are generallyforced to test undesirable and unpredictable PMD states. Often, PMDemulators include ten or more birefringent sections.

[0007] A PMD generator can also be incorporated into a specializedtelecommunications sub-system called a PMD compensator. PMD compensatorsare used to mitigate the deleterious effects of PMD imparted on anoptical data signal transmitted through an optical fiber. In contrast toPMD emulators, PMD compensators generally include only one or twobirefringent sections, but such a small number of sections greatlylimits the range of achievable PMD states. In order to achieve a greateroperating range, it may be desirable to use PMD compensators thatinclude more than two birefringent generator sections. Unfortunately,PMD spectra generated with more than two sections are difficult tocontrol, subject to misalignment, and are frequency dependent.

[0008] The number of birefringent sections is known to at leastpartially determine how much structure exists in the resultant PMDmagnitude spectrum. If one were to take the Fourier transform of anexemplar PMD-magnitude spectrum artificially generated by severalbirefringent sections, several Fourier component frequencies would beevident. The number of sinusoidal Fourier components depends generallyon the number of birefringent sections. For example, one birefringentsection generates a PMD-magnitude spectrum that has only one Fouriercomponent, the average, or DC, component. Two birefringent sections alsogenerate a PMD spectrum whose magnitude also has only one Fouriercomponent, again the DC component. Each additional birefringent sectioncan generate multiple sinusoidal Fourier components that appear in theresultant PMD spectrum.

[0009] It is known that a concatenation of several birefringent sectionscan be used to synthesize a particular optical intensity spectrum. Forexample, in 1949 Evans, an astronomer, described a birefringent filterto improve solar observations (see, Evans “The Birefringent Filter,” J.Optical Soc. of America, Vol. 39, No. 3, at 229-242 (March, 1939))(hereinafter, “Evans”). Similarly, in 1961 Harris described ageneralized filter synthesis method using birefringent filters (see,Harris et al. “Optical Network Synthesis Using Birefringent Crystals,”J. Optical Soc. of America, Vol. 54, No. 10, at 1267-1279 (March, 1964))(hereinafter, “Harris”). In both cases, a multi-stage birefringentfilter was placed between two polarizers to generate an opticalintensity spectrum.

[0010] Bührer U.S. Pat. No. 4,987,567 (hereinafter, “Bührer”) describesan alternative device that includes a multi-stage birefringent filterbetween two polarization diversity stages. According to this design,optical power transmission was increased, albeit in the form of twooptical beams. Buhrer's design has been extended to optical interleavers(see, e.g., U.S. Pat. Nos. 6,301,046, 6,215,923, 6,212,313, and6,252,711).

[0011] Thus, Evans, Harris, and Bührer showed coherent birefringentfilters. As used herein, a coherent birefringent filter is one in whicheach of the birefringent elements exhibits an optical retardation thatis an integral multiple of a unit reference optical retardation, whichmust itself be an integral multiple of 2π.

[0012] Fourier analysis of the resultant optical intensity spectrumgenerated by such coherent birefringent filters can, in general, revealmultiple sinusoidal frequency components. Moreover, it is known that therelative phase between each periodic component can be fixed to zero. Afilter that exhibits multiple Fourier components having identical phasevalues, as transformed from an optical intensity spectrum, is referredto herein as a coherent filter. In general, a coherent optical filterexhibits high periodicity and high contrast ratio in its opticalintensity spectrum.

[0013] Unlike the optical filtering shown by Evans, Harris, and Bührer,PMD generation does not permit frequency-dependent loss nor does itpermit polarization-dependent loss. Unfortunately, the polarizers usedby Evans and Harris generally produce substantial frequency-dependentand polarization-dependent losses. Also, the polarization diversityscheme shown by Bührer causes frequency-dependent loss on at least oneof the output beams.

[0014] As mentioned above, it is known that PMD generators can beconstructed from concatenated polarization maintaining (hereinafter,“PM”) fibers. Rotation of fibers with respect to adjacent fibers can becoordinated in such a manner to generate various forms of PMD spectra.For example, I. T. Lima et al. reports a PMD emulator constructed with15 polarization maintaining fibers and intermediate rotatable connectors(see, Lima et al., “Polarization Mode Dispersion Emulator,” OFC 2000,Paper ThB4 (February 2000)). Alternatively, a PMD generator can beconstructed with a concatenation of birefringent crystals. In this case,rotation of adjacent birefringent crystals (or control of intermediatepolarization-transforming stages) can also be coordinated in such amanner to generate various forms of PMD spectra. For example, a PMDemulator can be constructed with 12 birefringent crystals (see, Damask,“A Programmable Polarization-Mode Dispersion Emulator for SystematicTesting of 10 Gb/s PMD Compensators,” OFC 2000, Paper ThB3 (March,2000)). None of the references, however, shows how to build a coherentPMD generator.

[0015] It would therefore be desirable to provide methods and apparatusfor controllably generating coherent PMD spectra.

[0016] It would also be desirable to provide methods and apparatus tofor generating coherent PMD spectra that coincide with the comb spectrumof a WDM optical communications system.

[0017] It would be further desirable to provide methods and apparatus tocontrol coherent artificial PMD generation to independently generatefirst and second order PMD.

SUMMARY OF THE INVENTION

[0018] It is therefore an object of the present invention to providemethods and apparatus for controllably generating coherent PMD spectra.

[0019] It is also an object of the present invention to provide methodsand apparatus for generating coherent PMD spectra that coincide with thecomb spectrum of a WDM optical communications system.

[0020] It is another object of the present invention to provide methodsand apparatus to control coherent artificial PMD generation toindependently generate first and second order PMD.

[0021] According to one aspect of the present invention, a coherent PMDgenerator for generating a coherent PMD spectrum is provided. Thegenerator includes at least four birefringent stages in optical series,thereby forming at least three pairs of adjacent stages. Each of thestages includes a harmonic differential group delay element and aphase-compensating element.

[0022] According to another aspect of the present invention, a colorlesscoherent PMD generator for generating a coherent PMD spectrum isprovided. In this case, the generator is not only coherent, but is alsomade colorless because the DGD elements are colorless and thephase-compensating elements are locked.

[0023] According to yet another aspect of the present invention, a PMDgenerator can be controlled to generate DGD and second order PMDindependently at at least one optical frequency by inducing polarizationmode-mixing between the pairs of stages.

[0024] Methods for using these PMD generators, including their use incompensators and emulators, are also provided.

BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The above and other objects and advantages of the invention willbe apparent upon consideration of the following detailed description,taken in conjunction with the accompanying drawings, in which likereference characters refer to like parts throughout, and in which:

[0026]FIG. 1 shows a schematic diagram of an illustrative coherent PMDgenerator according to this invention;

[0027]FIG. 2 shows illustrative Fourier transform spectra according tothis invention, which have been calculated as the square of the DGDspectrum that can be generated at the output of the generator shown inFIG. 1;

[0028]FIG. 3 shows a schematic diagram of an illustrative colorlesscoherent PMD generator according to this invention;

[0029]FIG. 4 shows a superposition of illustrative WDM power spectrumand DGD spectrum generated by the PMD generator shown in FIG. 3according to this invention, both as a function of optical frequency;

[0030]FIG. 5 shows a schematic block diagram of an illustrativeindependent first and second order PMD generator according to thisinvention;

[0031]FIG. 6 shows two sets of illustrative frequency-dependent spectrathat can be generated using, for example, the PMD generator of FIG. 5according to this invention;

[0032]FIG. 7 shows an illustrative contour plot of DGD and SOPMD valuesat an optical frequency for varying degrees of mode-mixing between thestages of the generator shown in FIG. 5;

[0033]FIG. 8 shows a schematic diagram of an illustrative colorless IFSOPMD generator according to this invention;

[0034]FIG. 9 shows a perspective view of a concatenation of fourbirefringent elements;

[0035]FIG. 10 shows a frequency-dependent DGD spectrum that can begenerated with the concatenation shown in FIG. 9;

[0036]FIG. 11 shows a Fourier analysis of the square of DGD spectrum ofFIG. 10;

[0037]FIG. 12 shows the four Fourier-component sinusoids with respectiveamplitudes and phases as plotted in optical frequency (excluding the DCcomponent) associated with the Fourier spectra of FIG. 11;

[0038]FIG. 13 shows an illustrative concatenation of four birefringentelements according to this invention, where each of these elements hasthe same DGD value;

[0039]FIG. 14 shows illustrative amplitude spectrum and phase spectrumassociated with a DGD spectrum (see FIG. 15) of the concatenation shownin FIG. 13 according to this invention;

[0040]FIG. 15 shows an illustrative DGD spectrum and Fourier-componentsinusoids according to this invention;

[0041]FIG. 16 shows illustrative concatenation of four like birefringentelements, as well as an optical input beam and an optical output beam,according to this invention;

[0042]FIG. 17 shows illustrative amplitude spectrum and phase spectrumassociated with a DGD spectrum (see FIG. 18) of the concatenation shownin FIG. 16 according to this invention;

[0043]FIG. 18 shows an illustrative DGD spectrum and Fourier-componentsinusoids according to this invention;

[0044]FIG. 19 shows a perspective view of an illustrative uniaxialbirefringent crystal cut as a parallelepiped with its extraordinary axisshown at the input;

[0045]FIG. 20 shows two beams having different wavelengths within thebirefringent crystal shown in FIG. 19 according to this invention;

[0046]FIG. 21 shows a magnified perspective view of the face of thecrystal of FIG. 19, including the orientations of the extraordinary andordinary axes;

[0047]FIG. 22 shows an illustrative apparatus including a birefringentcrystal located between two crossed polarizers, as well as threeassociated beat patterns;

[0048]FIG. 23 shows how the optical intensity varies through the lastpolarizer of FIG. 22 as a function of optical frequency;

[0049]FIG. 24 shows a perspective view of an illustrative birefringentcrystal having a length error;

[0050]FIG. 25 shows the effect of a crystal length error according tothis invention;

[0051]FIG. 26 compares the phase tolerance of an illustrativehigh-birefringent crystal and an illustrative low birefringent crystal1316 according to this invention;

[0052]FIG. 27 shows an illustrative high-birefringent crystal having alength error and a low-birefringent crystal according to this invention;

[0053]FIG. 28 shows two independent intensity spectra corresponding tothe crystals shown in FIG. 27 according to this invention;

[0054]FIG. 29 shows how the two spectra intensity shown in FIG. 28 addaccording to this invention;

[0055]FIG. 30 shows a schematic diagram of an illustrative four-stagecoherent PMD generator according to this invention;

[0056]FIG. 31 shows another illustrative coherent PMD generatoraccording to this invention that includes four birefringent stagesaccording to this invention;

[0057]FIG. 32 shows yet another illustrative PMD generator that is likethe generator shown in FIG. 30, except that electro-optic elements,rather than half-wave waveplates, are used to polarization mode-mix;

[0058]FIG. 33 shows non-colorless coherent PMD generator that includesfour birefringent stages according to this invention;

[0059]FIG. 34 shows an illustrative DGD spectrum and an illustrative WDMcomb channel spectrum, both as a function of optical frequency accordingto this invention;

[0060]FIG. 35 shows an illustrative colorless, coherent PMD generator,with an input optical beam and an output optical beam according to thisinvention;

[0061]FIG. 36 shows an illustrative DGD spectrum and an illustrative WDMcomb channel spectrum, both as a function of optical frequency,associated with the generator of FIG. 35 according to this invention;

[0062]FIG. 37 shows illustrative colorless and frequency-alignedcoherent PMD generator according to this invention;

[0063]FIG. 38 shows an illustrative DGD spectrum and an illustrative WDMcomb channel spectrum, both as a function of optical frequency,associated with the generator of FIG. 37 according to this invention;

[0064]FIG. 39 shows a chart that includes a set of constant DGD valuecontours that can be generated using a PMD generator according to thisinvention;

[0065]FIG. 40 shows a chart that includes a set of constant SOPMD valuecontours that can be generated using a PMD generator according to thisinvention;

[0066]FIG. 41 shows an illustrative chart that includes a set ofconstant DGD value contours and a set of constant SOPMD magnitude valuecontours within a boundary contour, all at an optical frequency,according to this invention; and

[0067]FIG. 42 shows another chart that includes two orthogonaltrajectories that can individually, or in combination, be used to form adither cycle.

DETAILED DESCRIPTION OF THE INVENTION

[0068] According to one aspect of this invention, a coherent PMDgenerator is provided. As used herein, a coherent PMD generator is anoptical device that generates a coherent differential group delay(hereinafter, “DGD”) spectrum: (1) that is harmonic and (2) whoseFourier components are in phase with one another. A harmonic DGDspectrum is a DGD spectrum that has Fourier component frequencies thathave a common Fourier-component frequency denominator (i.e., areintegral multiples of a unit Fourier-component frequency). It will beappreciated, therefore, that a DGD spectrum can be harmonic andincoherent, but a coherent DGD spectrum is always harmonic.

[0069] A coherent PMD generator according to another aspect of thisinvention can generate DGD spectra that exhibit high periodicity andhigh contrast ratios. The high-periodicity property can be used toadvantageously align the generated DGD spectrum to a comb of WDM signalsfor use in PMD emulators and compensators.

[0070] Moreover, according to yet another aspect of this invention, acoherent PMD generator can be used to independently generate and controlfirst and second order PMD.

[0071]FIG. 1 shows illustrative coherent PMD generator 100 according tothis invention. During operation, input optical beam 101 propagatessequentially through each of the optical elements within generator 100,producing output optical beam 102, which has imparted coherent PMDspectrum. Generator 100 includes a plurality of coherent birefringentstages 105, 106, and 107. Stage 105, for example, includes harmonic DGDelement 108 and respective phase compensator 109. Similarly, stages 106and 107 include harmonic DGD elements 110 and 112, and phasecompensators 111 and 113, respectively.

[0072] As used herein, DGD elements 108, 110, and 112 are harmonicbecause of their relationship to each other; that is, the relationshipbetween the DGD values of the DGD elements. Accordingly, a plurality ofDGD elements are considered harmonic when all of the respective DGDvalues are an integral multiple of a unit DGD value.

[0073] In addition to DGD elements 108, 110, . . . , and 112, each ofstages 105, 106, . . . , and 107 has respective phase compensatorelements 109, 111, . . . , and 113. The combination of a DGD element anda phase compensator element in a stage, however, does not necessarilyyield the target amount of retardation and, in general, has a residualoptical retardation. In stage 105, for example, the combination ofelements 108 and 109 generate a residual optical retardation. Thus,stages 105, 106, . . . , and 107 are coherent when: (1) DGD elements108, 110, . . . , and 112 are harmonic and (2) the respective residualoptical retardations are substantially the same. With respect to FIG. 1,then, PMD generator 100 is coherent when all DGD elements are harmonicand when all residual optical retardations are substantially the same.Although only three stages are shown in FIG. 1, it will be appreciatedthat the number of generation stages can be more or less than three.

[0074] A polarization mode-mixing element is located between any pair ofadjacent stages. Mixing element 120, for example, is located betweencoherent birefringent stages 105 and 106. Similarly, mixing element 121is located between stage 106 and a subsequent stage (not shown).Finally, mixing element 122 is located between a two stages, includingfinal stage 107.

[0075] A mode-mixing controller controls each mode-mixing element.Mode-mixing controller 123, for example, controls the degree ofpolarization mode-mixing generated by polarization mode-mixing element120. Likewise, mode-mixing controllers 124 and 125 control the degree ofpolarization mode-mixing generated by polarization mode-mixing elements121 and 122, respectively.

[0076]FIG. 2 shows illustrative Fourier transform spectra, which havebeen calculated as the square of the DGD spectrum generated at output102 of generator 100. The Fourier transform spectra include amplitudespectrum 131 and phase spectrum 132. Amplitude spectrum 131 is referredto as a harmonic amplitude spectrum because each of Fourier-componentfrequencies 140-145 is an integral multiple of unit Fourier-componentfrequency ω. For example, frequencies 141, 142, 143, and 144 areintegral multiples of unit frequency at 140 (i.e., 2, 3 , (N−1), and Ntimes frequency ω, where N is an integer. It will be appreciated that DCFourier-component frequency 145 is zero times unit Fourier-componentfrequency 140.

[0077] The amplitudes of Fourier-component frequencies 140 through 145are determined, in part, by the degree of polarization mode-mixinggenerated along coherent PMD generator 100. These amplitudes can bepositive or negative. The overall DGD spectrum generated at output 102of generator 100 is also coherent because the phase amplitudes of thephase components 148 are substantially zero. Thus all sinusoidal Fouriercomponents that form the DGD spectrum are aligned in phase and share anoptical frequency where all the sinusoids are either at a maximum or ata minimum.

[0078] According to another aspect of the present invention, FIG. 3shows illustrative colorless coherent PMD generator 200. In addition tobeing coherent, generator 200 is colorless. As used herein, the term“colorless” refers to the situation where the DGD value produced bygenerator 200 is substantially the same at any optical channel frequencyof a WDM comb spectrum. As used herein, the term “comb spectrum” refersto a spectrum that has channels that are equally spaced in frequency.

[0079] Generator 200 includes a plurality of colorless, coherentbirefringent stages 205, 206, . . . , and 207. Stage 205, for example,includes colorless harmonic DGD element 208 and phase-locking element209. Similarly, stages 206 and 207 include colorless harmonic DGDelements 210 and 212, and phase-locking elements 211 and 213,respectively. Elements 208, 210, . . . , and 212 are similar to 108,110, . . . , and 112, but are designed to have an additionalproperty—the multiplicative inverse of the unit DGD value issubstantially the same as channel spacing 260 along WDM comb spectrum251. For example, the multiplicative inverse of a 10 picosecond DGDvalue is 100 GHz, which is a common channel spacing for WDM systems.

[0080] As mentioned above, phase-locking elements 209, 211, . . . , and213 include all the properties of phase compensators 109, 111, . . . ,and 113. Moreover, the residual optical retardations of stages 205, 206,and 207 (resulting from the internal pairs of colorless DGD elements andphase-locking elements), are chosen to generate an appropriate PMDspectrum on output beam 202. The PMD spectrum can be tuned such that adefinable frequency of the generated DGD spectrum is aligned with adefinable frequency of the WDM comb spectrum. Alignment can mean, forexample, that center frequency 270 (located at the middle of flat DGDspectral segment 268) is aligned to WDM comb frequency 272. Colorless,coherent PMD generation has the advantage that the same apparatus can beused for PMD generation at any WDM optical channel frequency.

[0081] As shown in FIG. 3, colorless, coherent PMD generator 200includes polarization mode-mixing elements between adjacent birefringentstages, each of which is controlled by a mode-mixing controller. Mixingelement 220, for example, is located between stages 205 and 206.Similarly, mixing element 221 is located between stages 206 and asubsequent stage (not shown). Also, element 222 is located between twostages, including last stage 207.

[0082] Mode-mixing controllers control the degree of polarizationmode-mixing. For example, controller 223 controls the degree ofpolarization mode-mixing generated by polarization mode-mixing element220. Likewise, mode-mixing controllers 224 and 225 control the degreesof polarization mode-mixing generated by polarization mode-mixingelements 221 and 222, respectively.

[0083]FIG. 4 shows a superposition of illustrative WDM power spectrum251 and DGD spectrum 252 generated by generator 200, both as a functionof optical frequency. Free-spectral range 262 of colorless DGD spectrum252 is selected to be the same as channel spacing 260 along spectrum251. In this example, middle frequency 270 of flattened middle portion268 of DGD spectrum 252 is aligned with WDM channel center frequency272.

[0084] The portion of DGD spectrum 252 that rapidly changes (i.e., edgeportion 271 is, in this case, located between the WDM channels so thatno channel experiences the highly variable, and rapidly changing portionof the PMD spectrum. Because both spectra 251 and 252 are periodic andshare the same period, the same amount of PMD can be imparted to eachWDM channel.

[0085] According to another aspect of this invention, a PMD generatorcan be constructed that is capable of generating and controlling firstand second order PMD independently. FIG. 5 shows illustrativeindependent first and second order PMD Generator (hereinafter, “IFSO PMDgenerator”) 300. IFSO PMD generator 300 includes at least four (e.g.,four stages, eight stages, etc.) coherent birefringent stages 305, 306,307, and 308 and three intermediate polarization mode-mixing elements320, 322, and 324. Each of stages 305, 306, 307, and 308 includesharmonic DGD element 310, 312, 314, and 316, respectively. Preferably,the DGD values of these DGD elements are substantially the same.

[0086] IFSO PMD generator 300 also includes mode-mixing controllers 326and 328. In this embodiment, controller 326 controls elements 320 and324 and controller 328 only controls element 322. IFSO PMD generator 300has the remarkable property that first and second order PMD can begenerated and independently controlled at optical output 302 for aparticular comb of optical frequencies. As discussed more fully below,independent control of first and second order PMD generation has anumber of advantages when used in PMD emulators or compensators.

[0087] Coherent birefringent stages 305, 306, 307, and 308 includeharmonic DGD elements 310, 312, 314, and 316, respectively. The DGDvalues of these elements can be the substantially same. IFSO PMDgenerator 300 is like coherent PMD generator 100 in that the fourresidual optical retardations values generated in each stage is largelydetermined by pairs of harmonic DGD elements 310, 312, 314, and 316 andrespective phase compensator elements 311, 313, 315, and 317. In thisembodiment, these residual optical retardations are substantially thesame. As already discussed above, each of phase compensators 311, 313,315, and 317 is selected separately to compensate for phase errorspresent in its paired DGD element.

[0088]FIG. 6 shows two sets of illustrative frequency-dependent spectrathat can be generated using, for example, IFSO PMD generator 300. Upperset 404 is a series of DGD spectra and lower set 407 is a series ofmagnitude second order PMD (hereinafter, “SOPMD”) spectra. Each setincludes seven spectra as a function of optical frequency 405corresponding to seven degrees of mode-mixing determined by controllers326 and 328.

[0089] It will be appreciated that both upper and lower sets 404 and 407are periodic and have free-spectral range 408. Spectral center frequency410 corresponds to the maximum DGD value for generated DGD spectrum 404.It will be further appreciated that the DGD and SOPMD values atfrequency 410 can be determined for all degrees of mode-mixing, whichare controlled by controllers 326 and 328.

[0090] For example, FIG. 7 shows illustrative contour plot 500 of DGDand SOPMD values at frequency 410 for values 501 of mode-mixingcontroller 326 and values 502 of mode-mixing controller 328. As shown inFIG. 7, values 501 and 502 of controllers 326 and 328 lie within thespace defined by contours 505 and 506. Within this space, DGD and SOPMDvary monotonically between zero and a maximum. Contour set 508 showsvarious combinations of controller values 501 and 502 that maintain aparticular DGD value. Similarly, contour set 509 shows variouscombinations of controller values 501 and 502 that maintain a particularSOPMD value. Thus, IFSO PMD generator 300 can access any DGD/SOPMD statewithin the restricted space, and particularly, can either: (1) accessany DGD value with or without changing the concomitant SOPMD value, or(2) access any SOPMD value with or without changing the concomitant DGDvalue.

[0091] Alternatively, generator 300 can access any DGD/SOPMD state alongany predetermined trajectory. That trajectory can be, for example, aconstant DGD value trajectory, a constant SOPMD value trajectory, afixed rate of change of a DGD value trajectory, a fixed rate of changeof a SOPMD value trajectory, and any combination thereof.

[0092]FIG. 8 shows illustrative colorless IFSO PMD generator 600according to this invention. Generator 600 combines the technology usedto create colorless coherent PMD generator 200 and IFSG generator 300.Colorless IFSO PMD generator 600 can advantageously generate PMD onoutput beam 602 with: (1) independent control of first and second orderPMD and (2) the same PMD state for each channel of the WDM channel comb.A colorless IFSO PMD generator can be especially useful when used in aPMD compensator because only one generator is necessary to generate aselectable amount of first and second order PMD for every WDM channel.

[0093]FIG. 8 shows illustrative colorless IFSO PMD generator 600. IFSOPMD generator 600 includes at least four coherent birefringent stages605, 606, 607, and 608 and three intermediate polarization mode-mixingelements 620, 622, and 624. Each of stages 605, 606, 607, and 608includes a colorless harmonic DGD element/phase-locking element pair610, 611, 612, and 613, respectively.

[0094] Generator 600 also includes mode-mixing controllers 626 and 628.Like generator 300, controller 626 of generator 600 controls elements620 and 624 and controller 628 only controls element 622. IFSO PMDgenerator 600 can generate and independently control first and secondorder PMD for a particular comb of optical frequencies.

[0095] Generator 600 is like coherent PMD generators 100 and 300 in thatthe four residual optical retardations values generated in each stage islargely determined by the component DGD elements and phase elements, butin this case the residual optical retardations are substantially thesame and are phase-locked. As already discussed above, each of phasecompensation elements are selected to compensate for phase errorspresent in its paired DGD element. In particular, the phase-lockingelements are selected to generate four residual optical retardationvalues that are substantially the same at the output of eachcolorless-harmonic-DGD and phase-locking element pair, and such that thePMD spectrum at output beam 602 is appropriately aligned with the WDMcomb spectrum (see above). Also, controllers 326 and 328 are operatedsuch that independent first and second order PMD can be generated atoutput 602.

[0096] Thus, coherent PMD generator 100, colorless coherent PMDgenerator 200, IFSO PMD generator 300, and colorless, coherent PMDgenerator 600 all impart a controllable amount of coherent PMD onto anoutput beam. The following detailed description is divided into threeparts: coherent PMD generation, colorless PMD generation, andindependent first and second order PMD control.

[0097] Coherent PMD Generation

[0098] As mentioned above, PMD is an optical property that can begenerated by a concatenation of two or more birefringent elements. Suchconcatenations are known, in general, to generate PMDfrequency-dependent spectra. A PMD spectrum includes two components: thepolarization state of its Principal State of Polarization, or “PSP”, asrepresented in three-dimensional Stokes' space; and the DGD betweensignals aligned along the two orthogonal PSPs, as represented by apositive-definite magnitude in units of time. It is well known that DGDis just the magnitude of PMD of the concatenation.

[0099]FIG. 9 shows a perspective view of a concatenation of fourbirefringent elements 701-704, optical input beam 705 and optical outputbeam 706. Without loss of generality, and for purposes of illustrationonly, birefringent elements 701, 702, 703, and 704 will be considered asuniaxial birefringent. A birefringent element is a dielectric mediumthat exhibits more than one index of refraction. A uniaxial birefringentmedium can be characterized by two ordinary refractive indices and oneextraordinary refractive index, where each refractive index lies alongone of three mutually orthogonal axes of the birefringent medium. Incontrast, a biaxial birefringent medium is generally characterized bythree different refractive indices, where each refractive index liesalong one of the three mutually orthogonal axes. The birefringence of auniaxial birefringent medium is the difference between the extraordinaryand ordinary refractive indices.

[0100] The different lengths of birefringent elements 701, 702, 703, and704 illustrate that the elements can have different DGD values. It iswell known that the DGD value of a single birefringent element is theproduct of its birefringence and its length, divided by the speed oflight. Birefringent elements 701-704 have DGD values τ₁, τ₂, τ₃, and τ₄and residual optical retardations φ₁, φ₂, φ₃, and φ₄, respectively.

[0101] Representative extraordinary axes 710, 711, 712, and 713 areshown in the faces of the respective birefringent elements andillustrate one set of possible relative orientations. Polarizationmode-mixing occurs at the interface between adjacent elements.Polarization mode-mixing can be zero when the extraordinary axes betweentwo adjacent elements are parallel or perpendicular. Polarizationmode-mixing can be maximized when the two extraordinary axes between twoadjacent elements are at a 45 degree angle. Thus, selection ofappropriate DGD values for the individual birefringent elements andcontrol of the degree of polarization mode-mixing between these elementscan be used to controllably generate PMD.

[0102]FIG. 10 shows illustrative frequency-dependent DGD spectrum 721.Inspection of spectrum 721 reveals that it exhibits oscillations and isperiodic. By definition, DGD is the square-root of the determinant ofthe frequency-derivative of a unitary transformation matrixcorresponding to a birefringent concatenation. The spectrum of thefrequency-derivative of the unitary transformation matrix is thereforerelated to the square of the DGD spectrum. It will be furtherappreciated that a Fourier analysis of any DGD spectrum squared isdirectly representative of the number of birefringent elements and therespective DGD magnitudes and residual optical retardations of thoseelements.

[0103]FIG. 11 shows an illustrative Fourier analysis of the square ofDGD spectrum 721. As with any Fourier analysis, there is an amplitudeand phase associated with each Fourier component. Amplitude spectrum 725and phase spectrum 726 are plotted as a function of Fourier-componentfrequency. As shown in FIG. 11, and in general, a four-stagebirefringent concatenation yields four Fourier-component frequencies:individual frequencies 730 and 731, difference frequency 732, and sumfrequency 733. Individual frequencies 730 and 731 are the inverse of theDGD value from birefringent elements 702 and 703, respectively. That is,birefringent elements 701 and 704, located on either end of theconcatenation, do not contribute to the Fourier-component spectrum. ωdenotes Fourier-component frequency, and ω=1/96 .

[0104] Additionally, DC Fourier-component frequency 734 represents theaverage magnitude of the DGD-spectrum squared. Thus, in general, anN-stage birefringent concatenation generates Fourier-componentfrequencies associated with each birefringent element other than the endelements, and further generates sum and difference frequencies. TheFourier-component frequencies generated by a concatenation do not changeas the polarization mode-mixing change. The amplitudes and phases of theFourier-components, however, do change.

[0105]FIG. 12 shows the four Fourier-component sinusoids with respectiveamplitudes and phases as plotted in optical frequency (excluding DCcomponent 730) associated with the Fourier spectra of FIG. 11. Thesquare-root of the sum of components 740 and the DC component yields DGDspectrum 721 of FIG. 10. The amplitudes and phases of spectra 725 and726 are used to calculate sinusoidal components 740. The overallresultant DGD spectrum, while periodic, appears irregular, exhibits along periodicity, and can exhibit a low contrast ratio of maximum tominimum DGD magnitude. Concatenation 700 does not, in general, generatecoherent PMD.

[0106]FIG. 13 shows concatenation 800 of four birefringent elements801-804, as well as optical input beam 805 and output beam 806. Theseelements have the same DGD values (i.e., τ₁=τ₂=τ₃=τ₄) but may havedifferent residual optical retardations φ₁, φ₂, φ₃, and φ₄.

[0107]FIG. 14 shows illustrative amplitude spectrum 810 and phasespectrum 811 associated with resultant DGD spectrum 830 (see FIG. 15) ofconcatenation 800. The individual Fourier-component frequencies aredegenerate at frequency ω because the DGD magnitudes of the two middlestages in 800 are the same. Sum Fourier-component frequency is 2ω, anddifference Fourier-component frequency is zero.

[0108] Thus, concatenation 800 is not coherent but Fourier-componentfrequencies ω and 2ω are harmonic because the frequencies are equal toan integral number of a unit frequency ω. Because residual opticalretardations φ₁,φ₂, φ₃, and φ₄ are not necessarily equal to each other,the Fourier-component phase values are, in general, different atfrequencies ω and 2ω.

[0109]FIG. 15 shows resultant DGD spectrum 830, Fourier-componentsinusoid 831 (Fourier-component frequency ω) and Fourier-componentsinusoid 832 (Fourier-component frequency 2ω). Because theFourier-component phase values are different for each Fourier-componentfrequency, optical frequency 835, at which Fourier-component sinusoid831 reaches maximum 834, and optical frequency 839, at whichFourier-component sinusoid 832 reaches minimum 838, are not the same.Thus, a concatenation having the same DGD for each stage yields aharmonic Fourier-component spectrum, but does not necessarily yield acoherent DGD spectrum.

[0110]FIG. 16 shows concatenation 900 of four like birefringent elements901-904 and optical input beam 905 and output beam 906. Each of elements901-904 has the same DGD value τ and the same residual opticalretardation φ. FIG. 17 shows illustrative amplitude spectrum 910 andphase spectrum 911 associated with DGD spectrum 930 (see FIG. 18) ofconcatenation 900. As with concatenation 800, the individualFourier-component frequencies are degenerate at frequency ω because theDGD magnitudes of middle stages 902 and 903 are the same. SumFourier-component frequency 2ω, and difference Fourier-componentfrequency is zero. Because the residual optical retardations for allelements in concatenation 900 are the same, the Fourier-component phasevalues are also the same at Fourier-component frequencies ω and 2ω.

[0111]FIG. 18 shows resultant DGD spectrum 930, Fourier-componentsinusoid 931 (Fourier-component frequency ω and Fourier-componentsinusoid 932 (Fourier-component frequency 2ω). Because theFourier-component phase values are the same, optical frequency 938, atwhich Fourier-component sinusoid 931 reaches maximum 935 andFourier-component sinusoid 932 reaches minimum 936, is the same. Thus,in this special case, the resultant DGD spectrum is coherent because allof constituent Fourier-component sinusoids are in phase.

[0112] It will be recognized the a concatenation according to thisinvention can be coherent if (1) each of the elements has a DGD valuethat is substantially an integral multiple of a unit DGD value and (2)each residual optical retardation divided by its respective DGD value issubstantially the same for all elements.

[0113] For example, the four stages could have τ, 2τ, 4τ, and 8τ as DGDvalues. The resultant Fourier-component frequencies are harmonic becauseeach frequency is an integral number times a base frequency. Theperiodicity of the resultant DGD spectrum is, in general, different fromthe case where all DGD values are the same, but the property ofcoherence can be retained when all the Fourier-component phases align.

[0114] Stable birefringent elements should be used when constructingcoherent PMD generators. Birefringent elements that can be used inaccordance with this invention include birefringent crystals, such asyttrium ortho-vanadate (YVO₄), rutile, lithium niobate (LiNbO₃), mica,and crystalline quartz. High-birefringent crystals are birefringentcrystals that have a relatively high birefringence with respect toanother crystal. However, certain birefringent crystals are nominallyreferred to as high-birefringent crystals, such as YVO₄ and rutile, evenwithout reference to another crystal. In contrast, mica and crystallinequartz, for example, are often referred to as low-birefringent crystals.

[0115]FIG. 19 shows illustrative uniaxial birefringent crystal 1020 cutas a parallelepiped with its extraordinary axis (“e-axis”) shown at face1022 of the input. It is known that within any dielectric medium, suchas a birefringent crystal, the wavelength of an optical beam isshortened from the corresponding free-space wavelength by the value ofthe refractive index that the beam experiences. The refractive indexthat the beam experiences depends, at least partially, on thepolarization state of the beam. If the polarization state has acomponent that is aligned with the extraordinary axis of the crystal,that component experiences the extraordinary refractive index. The sameapplies for polarization components aligned with an ordinary axis.

[0116] It is also known that the velocity of an optical beam depends onthe refractive index that the beam experiences. Because of thisdependence, there are two distinct velocities possible within a uniaxialbirefringent material. Thus, a polarization component that is alignedwith the extraordinary axis travels at a different velocity from apolarization component aligned with one of the ordinary axes. Ingeneral, an arbitrary polarization state that enters a uniaxialbirefringent medium is resolved into two distinct beams, each having alinear orthogonal polarization state, each state being aligned withinternal crystalline axes, and each beam having distinct velocities.

[0117] For example, if a uniaxial birefringent crystal has ordinary andextraordinary refractive indices of 2.0 and 2.2, respectively, such acrystal is a positive uniaxial crystal. In this case, the wavelength ofan optical beam having its polarization state aligned with one of theordinary axes is shortened within the crystal by a factor of 2.0 whencompared to the wavelength of the optical beam traveling in free space.Similarly, the wavelength of another beam having its polarization statealigned with the extraordinary axis is shortened by a factor of 2.2 whencompared to its wavelength in free space.

[0118]FIG. 20 shows two beams having wavelengths λ_(e) and λ_(o) withincrystal 1020. The beam associated with wavelength λ_(e) has itspolarization component aligned along the extraordinary crystalline axiswhile the beam associated with longer wavelength λ_(o) has itspolarization component aligned along one of the ordinary axes. Bothbeams commence propagation at input plane 1025, which corresponds toface 1022, but because their wavelengths differ, the separation betweenpeaks of different beams increases and decreases during propagationthrough crystal 1020. Thus, the wave on one axis propagates in and outof phase with the wave on the other axis. FIG. 21 shows a magnifiedperspective view of crystal face 1022, including the orientations of theextraordinary and ordinary axes.

[0119] Optical polarization retardation, sometimes simply referred to asretardation, is a measure of phase slip between two polarizationcomponent beams. When two orthogonally polarized beams are in phase, theretardation is zero. When the same beams slip by one full wave, theretardation is 2π. Similarly, when the same beams slip by two fullwaves, the retardation is 4π. Retardation value is typically referred toin modulo 2π. Thus, any number of integral full wave slips correspondsto zero retardation. Optical retardation is thus better used as ameasure of the fractional slip in phase between two component opticalbeams. For example, a half-wave phase slip corresponds to a retardationof π.

[0120] The birefringent beat length, which is another measure ofbirefringence, is the physical length that corresponds to 2πretardation. Thus, the birefringent beat length is the free-spaceoptical wavelength at a given optical frequency divided by thebirefringence of the crystal. If, for example, the birefringence of acrystal is 2.2−2.0=0.2, then, for a free-space wavelength of 1.5microns, the birefringent beat length is 1.5/0.2 microns, or 7.5microns.

[0121]FIG. 22 shows illustrative apparatus including birefringentcrystal 1120 located between two crossed polarizers 1122 and 1124. Afterpolarizer 1122 linearly polarizes input beam 1123, the beam propagatesthrough birefringent crystal 1120 and is analyzed by crossed polarizer1124. Below the apparatus, FIG. 22 also shows beat patterns 1126, 1128,and 1130. Each beat pattern illustrates the beat between orthogonalpolarization components of an optical beam through the crystal. The beatlengths of beat patterns 1126, 1128, and 1130 correspond to thebirefringent beat lengths at three different optical frequencies. Itwill be appreciated that the optical intensity does not periodicallyvary through the crystal and that beat patterns 1126, 1128, and 1130 aremerely for illustrative purposes.

[0122] Patterns 1126 and 1130 show that an integral number ofbirefringent beats can exist in a birefringent crystal. In contrast,pattern 1128 shows that there can be residual retardation (some fractionof a birefringent beat) remaining at the end of the crystal. Theresidual retardation corresponds to the optical retardation that remainsafter the integral number of birefringent beats is subtracted. Also, asdemonstrated by the dashed lines in FIG. 22, a higher number of beatswithin crystal 1120 occurs as the optical frequency of the beamincreases (i.e., the wavelength decreases).

[0123]FIG. 23 shows how the optical intensity varies through analyzer1124 as a function of optical frequency. When the beat pattern atcrystal termination (end face) 1125 is maximized (e.g., trace 1126), thetransmitted intensity 1142 through polarizer 1124 is minimized.Similarly, intensity 1144 is maximized at termination 1125 when the beatterminates at a minimum (e.g., trace 1128). Further increase of theoptical frequency can restore the maximized beat pattern andcorresponding minimized transmitted intensity 1146 (e.g., trace 1130),albeit with an additional beat along the crystal length.

[0124] Thus, as shown in FIG. 23, as the optical frequency changes, anoptical intensity through analyzer 1124 traces a periodic waveform. Thefrequency separation between points 1142 and 1146 is the free-spectralrange (hereinafter, “FSR”) of crystal 1120.

[0125] For example, the birefringence of a YVO₄ crystal at 1.55 micronsis approximately 0.214. The beat length is therefore 1.55/0.214˜7.25microns. A YVO₄ crystal that is 14.022 mm long generates an FSR of about100 GHz—a convenient telecommunications value. Thus, within this crystalthere are approximately 1935 birefringent beat lengths from input faceto output face.

[0126] It will be appreciated that the fabrication of any crystal willresult in some degree of length error. When a birefringent crystal hasan error in its length, the number of birefringent beats and theresidual retardation can change when compared to the same crystal havingzero error. FIG. 24 shows a perspective view of illustrativebirefringent crystal 1200 having length error 1202. In this case,crystal 1200 is shorter than predicted. Accordingly, output plane 1205is moved towards input plane 1203, which truncates birefringent beatpattern 1201 and reduces the residual retardation.

[0127]FIG. 25 shows the effect of a crystal length error. In particular,frequency response 1204, which corresponds to crystal 1200, is shiftedupward from predicted frequency response 1206 by frequency error 1208.It will be appreciated that if length error 1202 is small compared tothe integral number of birefringent beats multiplied by the birefringentbeat length, then free-spectral range 1207 is not substantially altered.

[0128] Because the intensity spectrum is periodic, a phase shift can bedefined as frequency error 1208 divided by free-spectral range 1207. Aphase error is that phase shift which is associated with a length error.A retardation error is the difference between the anticipatedretardation and that retardation realized due to a length error. A phaseerror of an optical spectrum directly correlates to a retardation errorwithin a birefringent crystal.

[0129] A particular difficulty with the fabrication of high-birefringentcrystals with precise phase control is the short beat length of suchcrystals. In the above example, the beat length was approximately 7.25microns. To fabricate a crystal that is 14.022 mm long to within 7.25micron precision requires the ability to measure the crystal length toapproximately one part in 2,000. However, a 7.25 micron error results innearly a 2π band of phase error.

[0130] To reduce the nearly 2π band of phase error to, for example, aπ/5 phase error band, the crystal length would have to be controlled towithin 0.725 microns. A 0.725 micron length tolerance is difficult tomeasure and difficult to achieve. As an illustration of the difficultyinvolved, most modern high-birefringent crystals are polished to withinapproximately +/−5.0 microns of the target length.

[0131] The use of a low-birefringent crystal in combination with ahigh-birefringent crystal can result in a lower overall phase error. Auseful low-birefringent crystal is crystalline quartz, which has abirefringence Δn˜0.0084. The An ratio between YVO₄ and quartz is about25:1. The birefringent beat length in crystalline quartz is thereforeabout 25 times longer than in YVO_(4.)

[0132]FIG. 26 compares the phase tolerance of high-birefringent crystal1312 (e.g., YVO₄) and low birefringent crystal 1316 (e.g., quartz).Beats 1314 within crystal 1312 have a relatively short beat length 1310and beats 1320 within crystal 1316 have a relatively long beat length1318. Thus, for the same crystal length tolerance during fabrication, acrystal with lower birefringence will have higher phase tolerance.

[0133] The combination of a high birefringent crystal with low phasetolerance and a low-birefringent crystal with high phase tolerance canresult in an overall system with high phase tolerance. As such, alow-birefringent crystal can be used as a phase compensator for a highlybirefringent crystal. Accordingly, a phase compensator can be one ormore low-birefringent crystals that substantially correct for the phaseerror of one or more high-birefringent different crystals.

[0134]FIG. 27 shows an illustrative high-birefringent crystal 1400(having length error 1401) and low-birefringent crystal 1402. Theextraordinary axis 1408 of crystal 1400 and extraordinary axis 1409 ofcrystal 1402 are preferably either parallel or perpendicular to oneanother.

[0135]FIG. 28 shows two independent intensity spectra 1410 and 1412corresponding to crystals 1400 and 1402, respectively. As shown, theslope of spectrum 1412 is negative, and when added to intensity spectrum1410, resultant spectrum 1414 (shown in FIG. 29) is effectively shiftedto the left (a lower frequency). Thus, retardation 1406 of crystal 1402adds to residual retardation 1404 of crystal 1400, such that the overallretardation is greater than that of high birefringent crystal 1400alone. Thus, composite intensity spectrum 1414 is effectively shifted tolower frequencies.

[0136]FIG. 30 illustrates four-stage coherent PMD generator 1500according to this invention, including input and output optical beams1501 and 1502. It will be appreciated, however, that the number ofstages is not limited to four.

[0137] Generator 1500 includes four birefringent stages 1508, 1520,1530, and 1540, each of which includes a high-birefringent element and arespective low-birefringent element that is selected to minimize theoptical retardation error of the high birefringent element (i.e., thestage). In this particular embodiment, the high birefringent elementshave substantially the same DGD values τ, but, in general, each elementcan have any DGD value that is an integral multiple of a unit DGD value.Stage 1508, for example, includes high-birefringent element 1510, whichhas length error 1514, and low birefringent element 1512. Length error1514 introduces an optical retardation error φ₁, which is compensated byselection of phase-compensating low birefringent element 1512 havingresidual retardation +φ₁.

[0138] It will be appreciated that a birefringence error of the materialthat makes up element 1510 rather than length error 1514 can alsointroduce residual retardation error φ₁. Likewise, residual retardation+φ₂ of element 1526 is mitigated by residual retardation −φ₂ Of selectedelement 1528; residual retardation +φ₃ of element 1536 is mitigated byresidual retardation −φ₃ of selected element 1538; and residualretardation −φ₄ of element 1536 is mitigated by residual retardation +φ₄of selected element 1538.

[0139] To avoid polarization mode-mixing between high birefringentelement 1510 and low birefringent element 1512, extraordinary axis 1516of element 1510 can be aligned substantially parallel to extraordinaryaxis 1518 of element 1512. Alternatively, extraordinary axes 1516 and1518 can be aligned substantially perpendicular to one another, as longas the residual optical retardation φ₁ of high birefringent element 1510remains mitigated.

[0140] Similarly, to avoid polarization mode-mixing between high and lowbirefringent elements, stage 1520 can have extraordinary axis 1522 ofhigh birefringent element 1526 and extraordinary axis 1524 of lowbirefringent element 1528 aligned in a substantially parallel orperpendicular fashion. Stage 1530 has extraordinary axis 1532 of highbirefringent element 1536 and extraordinary axis 1534 of lowbirefringent element 1538 aligned in a substantially parallel orperpendicular fashion. Furthermore, stage 1540 has extraordinary axis1542 of high birefringent element 1546 and extraordinary axis 1544 oflow birefringent element 1548 aligned in a substantially parallel orperpendicular fashion.

[0141] To induce a degree of polarization mode-mixing between adjacentstages, relative rotation of extraordinary axes is required. Forexample, extraordinary axis 1522 of stage 1520 can be rotated withrespect to extraordinary axis 1516 of stage 1508.

[0142] It will be appreciated that each stage of generator 1500 caninclude more than one high-birefringent element. When ahigh-birefringent element includes a single birefringent crystal, thecrystal can be chosen such that it generates any desired free-spectralrange. When two or more crystals are combined in a single stage, thecombination can be chosen to optimize one or more physical attributes,including, for example, the free-spectral range, the optical retardationtemperature coefficient, the thermal expansion coefficient, and anycombination thereof. For example, a high birefringent stage may beconstructed using a YVO₄ crystal and a LiNbO₃ crystal with extraordinaryaxes aligned. The length ratio of YVO₄ to LiNbO₃ crystals can beselected to minimize the temperature dependence of the opticalretardation for the combined crystals.

[0143]FIG. 31 shows another illustrative coherent PMD generator 1600according to this invention, which includes birefringent stages 1605,1606, 1607, and 1608. Generator 1600 is similar to generator 1500, withtwo exceptions. First, extraordinary axes 1615, 1616, 1617, and 1618 ofhigh birefringent elements 1610, 1611, 1612, and 1613, respectively, aresubstantially parallel (although they can also be substantiallyperpendicular). Second, half-wave waveplates 1620, 1621, and 1622 arelocated between stages 1605, 1606, 1607, and 1608, respectively, tocontrol mode-mixing.

[0144] The PMD added to output beam 1602 is controlled by rotation ofbirefringent axes 1625, 1626, and 1627 of half-wave waveplates 1620,1621, and 1622, respectively. Rotation is preferably about an axissubstantially parallel to beam 1601. In some ways, generator 1600 iseasier to build than generator 1500 because the birefringent axes of thefour stages are aligned and fixed in place while the half-wavewaveplates are rotated.

[0145]FIG. 32 shows yet another illustrative PMD generator 1650, whichis like generator 1600, except that electro-optic elements, rather thanhalf-wave waveplates, are used to polarization mode-mix. Electro-opticelement 1640 is located between birefringent stages 1605 and 1606,element 1641 is located between birefringent stages 1606 and 1607, andelement 1642 is located between birefringent stages 1607 and 1608. Likegenerator 1600, extraordinary axes 1615, 1616, 1617, and 1618 of highbirefringent elements 1610, 1611, 1612, and 1613 are substantiallyparallel.

[0146] During operation, mode-mixing can be induced by orientingelectro-optically induced birefringent (hereinafter, “principal”) axis1644 in a fashion that is neither parallel nor perpendicular toextraordinary axis 1615. For example, principal axis 1644 can be alignedat 45 degrees with respect to extraordinary axis 1615, such that bothaxes lie within a plane substantially perpendicular to beam 1601.Similarly, principal axis 1646 can be aligned at 45 degrees toextraordinary axis 1616, and principal axis 1648 can be aligned at 45degrees to extraordinary axis 1617.

[0147] Control voltages V1, V2, and V3 control the optical retardationsgenerated by electro-optic elements 1640, 1641, and 1642, respectively.The PMD generated at output 1602 by generator 1650 is controlled bycontrol voltages V₁, V₂, and V₃ of electro-optic elements 1640, 1641,and 1642. Generator 1650 can be operated at a higher speed thangenerator 1600 because a change of control voltages V₁, V₂, and V₃ canbe faster than rotation of half-wave waveplates 1620, 1621, and 1622.

[0148] For example, electro-optic elements 1640, 1641, and 1642 can bemade using LiNbO₃ crystals. The extraordinary axis of the LiNbO₃ crystalcan be cut such that the axis is substantially parallel to beam 1601.This cut eliminates additional optical retardation imparted to beam 1601when no voltage is applied. The x-axis of the LiNbO₃ crystal can be cut,and electrical contacts can be located on the crystal, such that theelectro-optically induced principal axis lies at a 45 degree angle withrespect to an applied electric field and lies in a plane that isperpendicular to the extraordinary axis.

[0149] It will be appreciated that although generators 1500, 1600, and1650 each include four birefringent stages having substantially zerooptical retardation, any number of stages can be used as long as: (1)the respective DGD values τ are either the same or an integral multipleof a unit DGD value and (2) the residual optical retardation value ofeach stage divided by its DGD value is substantially the same.

[0150] Colorless, Coherent PMD Generation

[0151] As used herein, a colorless, coherent PMD generator is capable ofgenerating the same PMD value on any optical channel along awavelength-division multiplexed comb of optical signals that are equallyspaced in frequency. Coherent PMD generators can be made colorless byselecting an appropriate unit DGD value and residual optical retardationvalue for each birefringent stage.

[0152] The unit DGD value is preferably the multiplicative inverse ofthe frequency separation between adjacent WDM channels. Also, theresidual optical retardation of each birefringent stage is preferablychosen to generate a PMD spectrum with a desirable alignment between anydefinable center frequency of the DGD spectrum and a definable frequencyof the WDM comb spectrum.

[0153]FIG. 33 shows non-colorless coherent PMD generator 1700, which inthis case includes four birefringent stages 1710, 1712, 1714, and 1716,although a different number of stages can be used. The DGD value τ_(a)of high birefringent stages 1720, 1722, 1724, and 1726 yield afree-spectral range of the generated DGD spectrum at output 1702. FIG.34 includes DGD spectrum 1754 and WDM comb channel spectrum 1752, bothas a function of optical frequency 1751. WDM spectrum 1752 has a channelseparation 1758 and DGD spectrum has a free-spectral range 1766. Whenτ_(a) is not a multiplicative inverse of channel spacing 1758, or someintegral multiple thereof, the spectral periodicities of DGD spectrum1754 and WDM comb spectrum 1752 do not match.

[0154] Accordingly, DGD value 1764 of spectrum 1754 can coincide atfrequency 1760 with power value 1762 at a maximum of WDM comb spectrum1752, but point 1768 of DGD spectrum 1754 does not correspond to anothermaximum along WDM comb spectrum 1752. Therefore, generator 1700 is notcolorless because the PMD generated at output 1702 is not the same forall channels along a WDM comb spectrum.

[0155]FIG. 35 shows illustrative colorless, coherent PMD generator 1800,with input optical beam 1801 and output optical beam 1802. As describedmore below, the PMD spectrum generated by generator 1800 isfrequency-shifted compared with generator 1700. Generator 1800 includesfour birefringent stages 1810, 1812, 1814, and 1816, but any number ofstages can be used according to this invention. High-birefringent stages1820, 1822, 1824, and 1826 have the same DGD value τ_(b), which yields agenerated DGD spectrum that is periodic and has a free-spectral range.DGD value τ_(b) can be the multiplicative inverse of channel separation1868, or any integral multiple thereof.

[0156]FIG. 36 shows the frequency-dependence of DGD spectrum 1854 andWDM comb channel spectrum 1852. Unlike free-spectral range 1766,free-spectral range 1866 of the DGD spectrum is substantially equal tochannel separation 1868 of the channel spectrum. Because theperiodicities of DGD spectrum 1854 and WDM comb spectrum 1852essentially the same, generator 1800 is colorless.

[0157] Although the periodicities of DGD spectrum 1854 and WDM combspectrum 1852 are essentially the same, their phase relationship may notbe desirable. In other words, a predetermined center frequency of theDGD spectrum and a predetermined frequency of the WDM comb spectrumcould have an undesirable frequency difference. For example, in FIG. 36,optical frequency 1861 (corresponding to DGD value 1864 of spectrum1854, which may be a desirable center frequency of the DGD spectrum) andfrequency 1860 (corresponding to maximum-power point 1862 along WDM combspectrum 1852) do not substantially coincide. The difference betweenoptical frequencies 1860 and 1861 (hereinafter, “frequency error” 1869)measures the spectral misalignment between spectra 1852 and 1854.

[0158] Frequency error 1869 is the result of optical retardation error−φ_(ε) present in birefringent stages 1820, 1822, 1824, and 1826 and isnot compensated with phase compensators 1830, 1832, 1834, and 1836.Frequency error 1869 divided by channel separation 1868 and multipliedby 2π yields a phase value, where the phase value is equal to theoptical retardation error −φ_(ε). Thus, generator 1800 is colorless, butthe frequency error between resultant DGD and WDM comb spectra can yieldan undesirable PMD spectrum alignment with the WDM channels.

[0159]FIG. 37 shows illustrative colorless and frequency-alignedcoherent PMD generator 1900, with input optical beam 1901 and outputoptical beam 1902. Generator 1900 includes four birefringent stages1910, 1912, 1914, and 1916, but it will be appreciated that any numberof stages can be used according to this invention. The four stages havethe same DGD value τ_(b), which yields a generated DGD spectrum that isperiodic and has a free-spectral range. As discussed above, DGD valueτ_(b) can be the multiplicative inverse of channel separation 1868, orany integral multiple thereof.

[0160]FIG. 38 shows illustrative DGD spectrum 1954 and WDM comb channelspectrum 1952, both as a function of optical frequency 1951, which canbe generated by generator 1900. As already discussed above,free-spectral range 1966 and period 1968 of spectra 1952 and 1954,respectively, are essentially the same by selecting a DGD value τ_(b) tobe the multiplicative inverse of channel separation 1968.

[0161] By selecting an appropriate optical residual retardation value φ₀of birefringent stages 1920, 1922, 1924, and 1926, alignment can beachieved between DGD value 1964 and power maximum 1962 at opticalfrequency 1960. It will be appreciated that residual optical retardationφ₀ of a stage (i.e., stage 1910, 1912, 1914, or 1916) can reside in ahigh birefringent element (i.e., element 1920, 1922, 1924, or 1926), inphase compensator elements (i.e., element 1930, 1932, 1934, and 1936),or some combination of elements. In accordance with this invention, whenthe residual optical retardation φ₀ is chosen such that the generatedDGD and the WDM comb spectra are desirably aligned (or “locked”), phasecompensator elements 1930, 1932, 1934, and 1936 are referred to asphase-locking element elements.

[0162] Thus, generator 1900 can generate a colorless and phase-lockedPMD spectrum. The colorless property is achieved by selecting DGD valueτ_(b) to be the multiplicative inverse of the channel spacing of a WDMcomb spectrum (or any integral multiple thereof) and by selecting theresidual optical retardation value φ₀ to achieve an appropriate phaserelationship—namely, one in which there is no discernable frequencyshift between generated DGD spectrum and the WDM comb.

[0163] Independent First- and Second-Order PMD Generation

[0164] According to one aspect of this invention, a coherent PMDspectrum can be generated using any number of birefringent stages havingharmonic DGD values and coherent residual retardation values, regardlessof the degree of polarization mode-mixing present between the stages.

[0165] According to another aspect of this invention, independentgeneration and control of first and second order PMD (at an opticalfrequency within each DGD spectral period) can be achieved when thegenerator includes four harmonic birefringent stages havingsubstantially the same residual optical retardations and is operated ina special way. First, control of the degree of polarization mode-mixingbetween first and second stages, and the degree of polarizationmode-mixing between the third and fourth stages is linked. Second,control of the degree of polarization mode-mixing between the second andthird stages is coordinated with respect to the other degrees ofmode-mixing. As discussed more fully below, control algorithms orlook-up tables can be used to coordinate the degrees of polarizationmode-mixing in a repeatable and reliable way.

[0166] As mentioned earlier, ISFO generator 300 can be used to generatefirst order PMD and second order PMD at optical frequency 410. Frequency410 can correspond to a maximum DGD value along any DGD spectrum, andparticularly at any optical frequency that is shifted from the maximumDGD value by an amount equal to an integer multiplied by thefree-spectral range. Thus, in the case of generator 300, controller 326is used to control mode-mixing elements 320 and 324, and controller 328controls mode-mixing elements 322. It will be appreciated that thedegree of mode-mixing between stages can also be controlled by varyingthe orientation of the stages themselves, thereby eliminated the needfor a physical element between the stages.

[0167]FIG. 39 shows chart 2000. The distances along the horizontal andvertical axes, which can be measured in degrees, can represent the anglebetween the extraordinary axes of adjacent birefringent stages, as shownin FIG. 9. More generally, the distances represent one-half of the anglesubtended on the Poincaré sphere due to mode-mixing between adjacentstages. The mode-mixing can be produced by physical rotation of thebirefringent stages themselves, insertion of waveplates (e.g., half-wavewaveplates) between the stages, and insertion of electro-optic elementsbetween the stages. When half-wave waveplates are used, the horizontaland vertical distances measure twice the angle of the waveplateextraordinary axis rotation. When electro-optic elements are used, thehorizontal and vertical distances measure one-half the opticalretardation imparted by the elements.

[0168] In particular, the distance along the horizontal axis(hereinafter, “PM2”) represents the amount of mode-mixing between stages306 and 307. Similarly, the distance along the vertical axis(hereinafter, “PM1”) represents the amount of mode-mixing between stages305 and 306, as well as stages 307 and 308.

[0169]FIG. 39 includes a set of constant DGD value contours that can begenerated using a PMD generator according to one aspect of thisinvention. Thus, each contour represents a set of PM1/PM2 combinationsthat will generate a predetermined DGD value at a particular opticalfrequency within the free-spectral range of the spectrum.

[0170] Indicated DGD values 0.5-4.0 on chart 2000 are normalized DGDvalues. The actual DGD value produced by generator 300 is equal to theproduct of the normalized DGD value indicated on chart 2000 and thebirefringent stage DGD value τ of any DGD element (e.g., element 310).For example, all PM1/PM2 combinations along contour 2006 generate anormalized DGD value of 2 (actual DGD value is equal to 2 times the DGDvalue of a birefringent stage). Similarly, all PM1/PM2 combinationsalong contour 2007 generate a normalized DGD value of 4, which in thiscase is just the point (0,0). Finally, PM1/PM2 combinations alongcontour 2008 generate normalized DGD value 0, where PM1=PM2−90 degrees.

[0171] Although not wishing to be bound by any particular theory, it isbelieved that the DGD contours (such as the contours of chart 2000) canbe determined by the following equation:

τ₃₀₀=4τ|cos (PM 1)×|cos (PM 2−PM 1)|

[0172] where τ₃₀₀ is the DGD value at optical frequency 410 produced bygenerator 300 and τ is the DGD value of a birefringent stage.

[0173]FIG. 40 includes a set of constant SOPMD value contours that canbe generated using a PMD generator according to one aspect of thisinvention. Thus, each contour represents a set of PM1/PM2 combinationsthat will generate a predetermined SOPMD value at a particular opticalfrequency within the free-spectral range of the spectrum.

[0174] Indicated SOPMD values 0-16 on chart 2010 are normalized SOPMDvalues. Thus, the actual SOPMD value produced by generator 300 is equalto the product of the normalized SOPMD value indicated on chart 2010 andthe birefringent stage DGD value τ of any DGD element (e.g., element310). A contour includes a set of PM1/PM2 combinations that generate thesame SOPMD magnitude at optical frequency 410. For example, all PM1/PM2combinations along contour 2014 generate a normalized SOPMD magnitude of2.

[0175] Darkened boundary contour 2012 includes a special set of PM1/PM2combinations because all SOPMD magnitude contours within boundarycontour 2012 change monotonically. Also, boundary 2012 covers the fullnormalized SOPMD magnitude range, from 0-16. For example, contour 2014shows a set of PM1/PM2 combinations that generate a normalized SOPMDmagnitude of 2. Contour 2016 shows another set of PM1/PM2 combinationsthat generate a normalized SOPMD magnitude of 2. However, no PM1/PM2combination within boundary contour 2012 produce contours having thesame SOPMD magnitude value. When the SOPMD contour is monotonic, it cansimplify the control of an IFSO PMD generator within a feedback controlloop, such as a PMD compensator.

[0176] The SOPMD contours on chart 2010 were obtained from the numericalmodeling of generator 300. Alternatively, and not wishing to be bound byany particular theory, it is believed that the SOPMD contours can alsobe analytically determined at optical frequency 410 as follows:

|τ_(w)|=τ²×{square root}(τ_(1w) ²+τ_(2w) ²+τ_(3w) ²),

where

τ_(1w)=−(1+cos (PM 1))(1+cos (PM 2−PM 1)),

τ_(2w)=(1+cos (PM 1))(sin 2(PM 2−PM 1)+sin (PM 2−PM 1))−2 sin (PM 1),and

τ_(3w)=0

[0177] and where |τ_(w)| is the SOPMD value at optical frequency 410produced by generator 300 and T is the DGD value of a birefringentstage.

[0178]FIG. 41 shows illustrative chart 2020, which includes a set ofconstant DGD value contours and a set of constant SOPMD magnitude valuecontours within boundary contour 2012 at an optical frequency. Allcontours shown in FIGS. 7 and 39-42 are normalized. As mentioned above,both DGD and SOPMD magnitude contours are monotonic within the boundaryregion. In accordance with this invention, these contours can be used tocontrol first order PMD and second order PMD independently from oneanother at an optical frequency.

[0179]FIG. 41 shows an example trajectory from PM1/PM2 combination a toPM1/PM2 combination d, via combinations b and c. In this case, onlyfirst order PMD or second order PMD varies at any given time. It will beappreciated that although all PM1 and PM2 values have been selected toremain within boundary contour 2012 to ensure monotonicity, trajectoriesthat extend outside boundary contour are possible.

[0180] Trajectory 2022, which extends from combination a to combinationb, follows a contour of constant SOPMD magnitude. Accordingly, the DGDvalue along trajectory 2022 decreases toward combination b while theSOPMD magnitude is constant. Trajectory 2024, which extends fromcombination b to combination c, follows a contour of constant DGD value.Accordingly, the SOPMD magnitude along trajectory 2024 increases towardscombination c while the DGD value is constant. Finally, trajectory 2026,which extends from combination c to combination d, again follows acontour of constant SOPMD magnitude. Accordingly, the DGD value alongtrajectory 2026 decreases toward combination d while the SOPMD magnitudeis fixed.

[0181] Thus, chart 2020 shows one of many possible examples where thePMD which is generated at output 302 is controlled to change DGD with nocorresponding change to SOPMD, and likewise is controlled to changeSOPMD with no corresponding change to DGD.

[0182]FIG. 42 shows chart 2030, which includes two orthogonaltrajectories 2032 and 2034 that can individually, or in combination, beused to form a dither cycle. Dithering is a well-known technique fordetermining the sensitivity of a system's performance to a particulardithered parameter. In the case of an optical network, it is known thatthe quality of a transmitted pulse depends on first and second order PMDdifferently. Thus, it would be desirable to identify whether first orderPMD or second order PMD was responsible for any degradation in signalquality. Thus, according to another aspect of this invention, one canmonitor the response of dithering first order PMD individually, secondorder PMD individually, or a known combination of both orders.

[0183] For example, for one part of a dither cycle, trajectory 2032varies the output DGD value but not the SOPMD magnitude. For anotherpart of the dither cycle, trajectory 2034 varies the SOPMD magnitude butnot the DGD value. A distortion analyzer (not shown) can then be used tomeasure whether the output signal is more sensitive to the first order(i.e., DGD) dithering or the SOPMD dithering. Once a measurement ismade, an error signal can be generated for controlling, for example, theappropriate amounts of first and second order compensatory PMD.

[0184] Colorless IFSO PMD Generation

[0185] A colorless IFSO PMD generator is a combination of a coherent,colorless PMD generator and an IFSO PMD generator.

[0186] As described above, FIG. 8 shows illustrative colorless ISFOgenerator 600. Generator 600 includes four coherent birefringent stages605, 606, 607, and 608, each of which has colorless, harmonic DGDelement and phase-locking element pairs 610, 611, 612, and 613,respectively. The DGD values of the four DGD elements are substantiallythe same, and the DGD value is chosen to be the multiplicative inverseof the channel spacing along a WDM comb spectrum. As a result, theperiod of resultant DGD spectrum (e.g., period 1966 of spectrum 1954)matches the channel separation of a WDM comb spectrum (e.g., channelseparation 1968 of spectrum 1952).

[0187] The four phase-locking elements are selected to generate fourresidual optical retardation values that are substantially the same atthe output of each colorless-harmonic-DGD and phase-locking elementpair, and further where the PMD spectrum on output beam 602 is desirablyaligned to a WDM comb. For example, that the frequency corresponding toa maximum along a generated DGD spectrum is substantially the same as afrequency corresponding to a maximum along a WDM comb spectrum.

[0188] Polarization mode-mixing elements 620, 622, and 624, are locatedbetween stages 605 and 606, 606 and 607, and 607 and 608, respectively.Controller 326 controls polarization mode-mixing elements 620 and 624.Controller 628 controls polarization mode-mixing element 622. Charts2000, 2010, and 500 (for example) show combinations of first and secondorder mode-mixing values for mode-mixing elements 620 and 624, and 622,respectively, that produce contours of constant DGD and SOPMD.

[0189] Thus, methods and apparatus for coherent PMD generation,colorless coherent PMD generation, independent control of first andsecond order PMD, and colorless PMD generation having independentcontrol of first and second order PMD are provided. One skilled in theart will appreciate that the present invention can be practiced by otherthan the described embodiments, which are presented for purposes ofillustration and not of limitation. For example, most of the PMDgenerators according to this invention can be constructed in a foldedgeometry, as taught by Damask U.S. Pat. application ser. No. 09/911,898,filed Jul. 24, 2001, which is hereby incorporated by reference in itsentirety. The present invention is limited only by the claims thatfollow.

What is claimed is:
 1. A coherent polarization mode dispersion (“PMD”)generator for generating a coherent PMD spectrum, wherein said generatorcomprises at least four birefringent stages in optical series, saidstages forming at least three pairs of adjacent stages, and wherein eachof said stages comprises a harmonic differential group delay (“DGD”)element and a phase-compensating element.
 2. The generator of claim 1wherein, for each stage, said phase-compensating element has anextraordinary axis and said DGD element has an extraordinary axis thatare oriented to avoid polarization mode-mixing within said each stage.3. The generator of claim 2 wherein, for said each stage, saidextraordinary axes have an orientation that is selected from a groupconsisting of substantially perpendicular and substantially parallel. 4.The generator of claim 1 wherein each of said stages has a residualoptical retardation that, when divided by a DGD value of its DGDelement, is substantially the same for each of said stages.
 5. Thegenerator of claim 1 wherein each of said stages has a total retardationthat is an integral multiple of π.
 6. The generator of claim 5 wherein,for each of said stages, said DGD element has a retardation error andsaid phase compensating element has a compensating retardation such thatsaid total retardation is said integral multiple of π.
 7. The generatorof claim 1 wherein said PMD spectrum comprises a DGD spectrum that hasFourier component frequencies that are in phase.
 8. The generator ofclaim 1 wherein said Fourier component frequencies have a commonFourier-component frequency denominator.
 9. The generator of claim 1wherein said PMD spectrum comprises a DGD spectrum that has sinusoidalFourier components that are all aligned in phase and share an opticalfrequency where all the sinusoidal components are either at a maximum orat a minimum.
 10. The generator of claim 1 wherein each of said DGDelements has a DGD value that is substantially equal to an integralmultiple of a unit DGD value.
 11. The generator of claim 10 wherein eachof said DGD elements has a DGD value is substantially the same.
 12. Thegenerator of claim 10 wherein each of said DGD elements has a differentDGD value.
 13. The generator of claim 10 wherein said unit DGD value issubstantially equal to a multiplicative inverse of a WDM channelspacing.
 14. The generator of claim 13 wherein said phase-compensatingelement is a locked phase-compensating element.
 15. The generator ofclaim 7 wherein said DGD spectrum is aligned to a comb of WDM signals.16. The generator of claim 1 wherein said generator is capable ofinducing a degree of polarization mode-mixing between at least one ofsaid pairs of stages.
 17. The generator of claim 16 wherein each of saidstages has a birefringent axis, such that when each of said stages isrotated, said birefringent axis is rotated.
 18. The generator of claim17 wherein each of said stages can be physically rotated about a beampropagation axis of said generator to induce polarization mode-mixingbetween at least one of said pairs of stages, but not within any of saidstages.
 19. The generator of claim 16 further comprising a polarizationmode-mixing element located between at least one of said pairs ofstages.
 20. The generator of claim 19 wherein each of said stages has abirefringent axis, and each of said pairs of stages includes a firststage and a second stage, wherein said first stage birefringent axis andsaid second stage birefringent axis is either substantially parallel orsubstantially perpendicular.
 21. The generator of claim 19 wherein saidmode-mixing element comprises at least one waveplate.
 22. The generatorof claim 21 wherein said at least one waveplate comprises a half-wavewaveplate.
 23. The generator of claim 22 wherein said half-wavewaveplate is rotatable about a beam propagation axis.
 24. The generatorof claim 19 further comprising at least one waveplate controller coupledto said at least one waveplate for controlling rotation of said at leastone waveplate.
 25. The generator of claim 19 wherein said mode-mixingelement comprises an electro-optic element.
 26. The generator of claim25 wherein said electro-optic element is located between a first pair ofsaid adjacent stages, said first pair comprising a first of said stagesand a second of said stages, wherein said electro-optic element has aprincipal axis and each of said first and second stages has abirefringent axis, and wherein said principal axis is not substantiallyparallel nor substantially perpendicular to either of said birefringentaxes of said first and second stages.
 27. The generator of claim 26wherein said principal axis and said birefringent axes of said first andsecond stages are at an angle that is about 45 degrees.
 28. Thegenerator of claim 16 further comprising at least one voltage source fordriving said electro-optic element.
 29. The generator of claim 16wherein said at least one pair of adjacent stages comprises every pairof said adjacent stages.
 30. The generator of claim 29 furthercomprising at least one mode-mixing element controller coupled to saidat least one mode-mixing element for controlling said mode-mixingelement.
 31. The generator of claim 29 wherein at least one of saidmode-mixing elements comprises an electro-optic element.
 32. Thegenerator of claim 1 wherein each of said DGD elements comprises abirefringent crystal selected from a group consisting of alpha bariumborate, yttrium ortho-vanadate, rutile, lithium niobate, mica,crystalline quartz, and any combination thereof.
 33. The generator ofclaim 1 wherein at least one of said DGD elements comprises a firstbirefringent crystal having an extraordinary axis and a secondbirefringent crystal having an extraordinary axis.
 34. The generator ofclaim 33 wherein said extraordinary axes have an orientation that isselected from a group consisting of substantially parallel andsubstantially perpendicular.
 35. The generator of claim 1 wherein atleast one of said DGD elements comprises a plurality of birefringentelements.
 36. The generator of claim 35 wherein said plurality ofbirefringent elements are chosen such that at least one physicalattribute is optimized, said attribute being selected from a groupconsisting of a free-spectral range, an optical retardation temperaturecoefficient, a thermal expansion coefficient, and any combinationthereof.
 37. A coherent polarization mode dispersion (“PMD”) generatorfor generating a coherent PMD spectrum, wherein said generator comprisesat least four birefringent stages in optical series, said stages formingat least three pairs of adjacent stages, and wherein each of said stagescomprises a colorless differential group delay (“DGD”) element and alocked phase-compensating element.
 38. The generator of claim 37wherein, for each stage, said phase-compensating element has anextraordinary axis and said DGD element has an extraordinary axis thatare oriented to avoid polarization mode-mixing within said each stage.39. The generator of claim 38 wherein, for said each stage, saidextraordinary axes have an orientation that is selected from a groupconsisting of substantially perpendicular and substantially parallel.40. The generator of claim 37 wherein each of said stages has a residualoptical retardation that, when divided by a DGD value of its DGDelement, is substantially the same for each of said stages.
 41. Thegenerator of claim 37 wherein each of said stages has a totalretardation that is an integral multiple of π.
 42. The generator ofclaim 41 wherein, for each of said stages, said DGD element has aretardation error and said phase compensating element has a compensatingretardation such that said total retardation is said integral multipleof π.
 43. The generator of claim 37 wherein said PMD spectrum comprisesa DGD spectrum that has Fourier component frequencies that are in phaseand further phase aligned with the phase of Fourier componentfrequencies of a WDM comb spectrum.
 44. The generator of claim 37wherein said Fourier component frequencies have a commonFourier-component frequency denominator is the same as a WDM channelspacing.
 45. The generator of claim 37 wherein said PMD spectrumcomprises a DGD spectrum that has sinusoidal Fourier components that areall aligned in phase and share an optical frequency where all thesinusoidal components are either at a maximum or at a minimum.
 46. Thegenerator of claim 37 wherein each of said DGD elements has a DGD valuethat is substantially equal to an integral multiple of a unit DGD value.47. The generator of claim 46 wherein each of said DGD elements has aDGD value is substantially the same.
 48. The generator of claim 46wherein each of said DGD elements has a different DGD value.
 49. Thegenerator of claim 46 wherein said unit DGD value is substantially equalto a multiplicative inverse of a WDM channel spacing.
 50. The generatorof claim 37 wherein said generator is capable of inducing a degree ofpolarization mode-mixing between at least one of said pairs of stages.51. The generator of claim 50 wherein each of said stages has abirefringent axis, such that when each of said stages is rotated, saidbirefringent axis is rotated.
 52. The generator of claim 51 wherein eachof said stages can be physically rotated about a beam propagation axisof said generator to induce polarization mode-mixing between at leastone of said pairs of stages, but not within any of said stages.
 53. Thegenerator of claim 50 further comprising a polarization mode-mixingelement located between at least one of said pairs of stages.
 54. Thegenerator of claim 53 wherein each of said stages has a birefringentaxis, and each of said pairs of stages includes a first stage and asecond stage, wherein said first stage birefringent axis and said secondstage birefringent axis is either substantially parallel orsubstantially perpendicular.
 55. The generator of claim 53 wherein saidmode-mixing element comprises at least one waveplate.
 56. The generatorof claim 55 wherein said at least one waveplate comprises a half-wavewaveplate.
 57. The generator of claim 56 wherein said half-wavewaveplate is rotatable about a beam propagation axis.
 58. The generatorof claim 53 further comprising at least one waveplate controller coupledto said at least one waveplate for controlling rotation of said at leastone waveplate.
 59. The generator of claim 53 wherein said mode-mixingelement comprises an electro-optic element.
 60. The generator of claim59 wherein said electro-optic element is located between a first pair ofsaid adjacent stages, said first pair comprising a first of said stagesand a second of said stages, wherein said electro-optic element has aprincipal axis and each of said first and second stages has abirefringent axis, and wherein said principal axis is not substantiallyparallel nor substantially perpendicular to either of said birefringentaxes of said first and second stages.
 61. The generator of claim 60wherein said principal axis and said birefringent axes of said first andsecond stages are at an angle that is about 45 degrees.
 62. Thegenerator of claim 50 further comprising at least one voltage source fordriving said electro-optic element.
 63. The generator of claim 50wherein said at least one pair of adjacent stages comprises every pairof said adjacent stages.
 64. The generator of claim 63 furthercomprising at least one mode-mixing element controller coupled to saidat least one mode-mixing element for controlling said mode-mixingelement.
 65. The generator of claim 63 wherein at least one of saidmode-mixing elements comprises an electro-optic element.
 66. Thegenerator of claim 37 wherein each of said DGD elements comprises abirefringent crystal selected from a group consisting of alpha bariumborate, yttrium ortho-vanadate, rutile, lithium niobate, mica,crystalline quartz, and any combination thereof.
 67. The generator ofclaim 37 wherein at least one of said DGD elements comprises a firstbirefringent crystal having an extraordinary axis and a secondbirefringent crystal having an extraordinary axis.
 68. The generator ofclaim 67 wherein said extraordinary axes have an orientation that isselected from a group consisting of substantially parallel andsubstantially perpendicular.
 69. The generator of claim 37 wherein atleast one of said DGD elements comprises a plurality of birefringentelements.
 70. The generator of claim 69 wherein said plurality ofbirefringent elements are chosen such that at least one physicalattribute is optimized, said attribute being selected from a groupconsisting of a free-spectral range, an optical retardation temperaturecoefficient, a thermal expansion coefficient, and any combinationthereof.
 71. The generator of claim 37 wherein said PMD spectrumcomprises a DGD spectrum having substantially the same DGD value at anycenter frequency of a WDM comb spectrum.
 72. The generator of claim 37wherein each of said stages has a total optical retardation equal to asum of a retardation of a DGD element and a retardation of a respectivelocked phase-compensating element, wherein each of said lockedphase-compensating elements can be tuned such that a first predeterminedfrequency of said DGD spectrum is aligned with a second predeterminedfrequency of a WDM comb spectrum.
 73. The generator of claim 72 whereinsaid first and second frequencies are either coincident or have apredetermined frequency difference.
 74. The generator of claim 37wherein a WDM spectrum has a plurality of center frequencies that areequally separated by a channel spacing, and wherein said DGD spectrumhas a periodic substantially flattened middle portion that issubstantially aligned with said center frequencies.
 75. The generator ofclaim 37 wherein a WDM spectrum has a plurality of center frequencies,and wherein said PMD spectrum comprises a frequency-dependent DGDspectrum having periodic stationary points that are substantiallycoincident with said center frequencies.
 76. A coherent polarizationmode dispersion (“PMD”) generator for generating a coherent PMDspectrum, wherein said generator comprises at least four birefringentstages in optical series, said stages forming at least three pairs ofadjacent stages, and wherein each of said stages comprises a harmonicdifferential group delay (“DGD”) element and a phase-compensatingelement, wherein said generator can be controlled to generate DGD andsecond order PMD (“SOPMD”) independently at at least one opticalfrequency by inducing polarization mode-mixing between said pairs ofstages.
 77. The generator of claim 76 wherein, for each stage, saidphase-compensating element has an extraordinary axis and said DGDelement has an extraordinary axis that are oriented to avoidpolarization mode-mixing within said each stage.
 78. The generator ofclaim 77 wherein, for said each stage, said extraordinary axes have anorientation that is selected from a group consisting of substantiallyperpendicular and substantially parallel.
 79. The generator of claim 76wherein each of said stages has a residual optical retardation that,when divided by a DGD value of its DGD element, is substantially thesame for each of said stages.
 80. The generator of claim 76 wherein eachof said stages has a total retardation that is an integral multiple ofπ.
 81. The generator of claim 80 wherein, for each of said stages, saidDGD element has a retardation error and said phase compensating elementhas a compensating retardation such that said total retardation is saidintegral multiple of π.
 82. The generator of claim 76 wherein said PMDspectrum comprises a DGD spectrum that has Fourier component frequenciesthat are in phase.
 83. The generator of claim 76 wherein said Fouriercomponent frequencies have a common Fourier-component frequencydenominator.
 84. The generator of claim 76 wherein said PMD spectrumcomprises a DGD spectrum that has sinusoidal Fourier components that areall aligned in phase and share an optical frequency where all thesinusoidal components are either at a maximum or at a minimum.
 85. Thegenerator of claim 76 wherein each of said DGD elements has a DGD valuethat is substantially equal to an integral multiple of a unit DGD value.86. The generator of claim 85 wherein each of said DGD elements has aDGD value is substantially the same.
 87. The generator of claim 85wherein each of said DGD elements has a different DGD value.
 88. Thegenerator of claim 85 wherein said unit DGD value is substantially equalto a multiplicative inverse of a WDM channel spacing.
 89. The generatorof claim 88 wherein said phase-compensating element is a lockedphase-compensating element.
 90. The generator of claim 82 wherein saidDGD spectrum is aligned to a comb of WDM signals.
 91. The generator ofclaim 76 wherein said generator is capable of inducing a degree ofpolarization mode-mixing between at least one of said pairs of stages.92. The generator of claim 91 wherein each of said stages has abirefringent axis, such that when each of said stages is rotated, saidbirefringent axis is rotated.
 93. The generator of claim 92 wherein eachof said stages can be physically rotated about a beam propagation axisof said generator to induce polarization mode-mixing between at leastone of said pairs of stages, but not within any of said stages.
 94. Thegenerator of claim 91 further comprising a polarization mode-mixingelement located between at least one of said pairs of stages.
 95. Thegenerator of claim 94 wherein each of said stages has a birefringentaxis, and each of said pairs of stages includes a first stage and asecond stage, wherein said first stage birefringent axis and said secondstage birefringent axis is either substantially parallel orsubstantially perpendicular.
 96. The generator of claim 94 wherein saidmode-mixing element comprises at least one waveplate.
 97. The generatorof claim 96 wherein said at least one waveplate comprises a half-wavewaveplate.
 98. The generator of claim 97 wherein said half-wavewaveplate is rotatable about a beam propagation axis.
 99. The generatorof claim 94 further comprising at least one waveplate controller coupledto said at least one waveplate for controlling rotation of said at leastone waveplate.
 100. The generator of claim 94 wherein said mode-mixingelement comprises an electro-optic element.
 101. The generator of claim100 wherein said electro-optic element is located between a first pairof said adjacent stages, said first pair comprising a first of saidstages and a second of said stages, wherein said electro-optic elementhas a principal axis and each of said first and second stages has abirefringent axis, and wherein said principal axis is not substantiallyparallel nor substantially perpendicular to either of said birefringentaxes of said first and second stages.
 102. The generator of claim 101wherein said principal axis and said birefringent axes of said first andsecond stages are at an angle that is about 45 degrees.
 103. Thegenerator of claim 91 further comprising at least one voltage source fordriving said electro-optic element.
 104. The generator of claim 91wherein said at least one pair of adjacent stages comprises every pairof said adjacent stages.
 105. The generator of claim 104 furthercomprising at least one mode-mixing element controller coupled to saidat least one mode-mixing element for controlling said mode-mixingelement.
 106. The generator of claim 91 wherein at least one of saidmode-mixing elements comprises an electro-optic element.
 107. Thegenerator of claim 76 wherein each of said DGD elements comprises abirefringent crystal selected from a group consisting of alpha bariumborate, yttrium ortho-vanadate, rutile, lithium niobate, mica,crystalline quartz, and any combination thereof.
 108. The generator ofclaim 76 wherein at least one of said DGD elements comprises a firstbirefringent crystal having an extraordinary axis and a secondbirefringent crystal having an extraordinary axis.
 109. The generator ofclaim 108 wherein said extraordinary axes have an orientation that isselected from a group consisting of substantially parallel andsubstantially perpendicular.
 110. The generator of claim 76 wherein atleast one of said DGD elements comprises a plurality of birefringentelements.
 111. The generator of claim 110 wherein said plurality ofbirefringent elements are chosen such that at least one physicalattribute is optimized, said attribute being selected from a groupconsisting of a free-spectral range, an optical retardation temperaturecoefficient, a thermal expansion coefficient, and any combinationthereof.
 112. The generator of claim 76 wherein said generator can becontrolled to generate differential group delay (“DGD”) and second orderPMD (“SOPMD”) independently at at least one optical frequency byinducing polarization mode-mixing between said pairs of stages.
 113. Thegenerator of claim 112 wherein each of said DGD elements has a DGD valuethat is substantially the same.
 114. The generator of claim 112 whereinsaid PMD generator is restricted to operate along a predeterminedDGD/SOPMD trajectory.
 115. The generator of claim 114 wherein saidpredetermined trajectory is selected from a group consisting of aconstant DGD trajectory, a constant SOPMD trajectory, a fixed rate ofchange of a DGD trajectory, a fixed rate of change of a SOPMDtrajectory, and any combination thereof.
 116. The generator of claim 115wherein each of said DGD and SOPMD at said optical frequency can varywithin a DGD/SOPMD space in which said DGD and said SOPMD only varymonotonically between zero and a respective maximum.
 117. The generatorof claim 76 wherein said at least one optical frequency is a pluralityof optical frequencies.
 118. The generator of claim 117 wherein saidindependent control of DGD and second order PMD can be simultaneous.119. The generator of claim 117 wherein said plurality of opticalfrequencies is a set of WDM channel frequencies, and wherein said firstand second amounts are substantially the same at each of said channelfrequencies.
 120. The generator of claim 76 wherein said at least fourstages comprises four stages.
 121. The generator of claim 120 whereinsaid four stages comprises: a first stage having an input and an output;a second stage having an input and an output, wherein said second stageinput is optically coupled to said first stage output; a third stagehaving an input and an output, wherein said third stage input isoptically coupled to said second stage output; and a fourth stage havingan input and an output, wherein said fourth stage input is opticallycoupled to said third stage output.
 122. The generator of claim 121further comprising: a first polarization mode-mixing controller forcontrolling a first degree of mode-mixing between said first and secondstages and between said third and fourth stages by physical rotation ofat least one of said stages; and a second polarization mode-mixingcontroller for controlling a second degree of mode-mixing between saidsecond and third stages by physical rotation of at least one of saidsecond and third stages.
 123. The generator of claim 122 furthercomprising: a first polarization mode-mixing element between said firststage and said second stage; a second polarization mode-mixing elementbetween said second stage and said third stage; and a third polarizationmode-mixing element between said third stage and said fourth stage. 124.The generator of claim 123 further comprising: a first polarizationmode-mixing element controller coupled to said first mode-mixing elementand said third mode-mixing element for controlling a first degree ofmode-mixing between said first and second stages and between said thirdand fourth stages; and a second polarization mode-mixing elementcontroller coupled to said second mode-mixing element for controlling asecond degree of mode-mixing between said second and third stages. 125.The generator of claim 124 wherein said first controller and said secondcontroller are programmed to coordinate mode-mixing between saidadjacent pairs of stages to generate said first amount of DGD and saidsecond amount of SOPMD within a free-spectral range of said PMDspectrum.
 126. The generator of claim 125 wherein said controllers areprogrammed to vary said PMD spectrum by changing the first and seconddegrees of mode-mixing such that said first amount of DGD issubstantially fixed while said second amount varies.
 127. The generatorof claim 125 wherein said controllers are programmed to vary said PMDspectrum by changing the first and second degrees of mode-mixing suchthat said first amount of DGD changes at a predetermined rate while saidsecond amount varies.
 128. The generator of claim 125 wherein saidcontrollers are programmed to vary said PMD spectrum by changing thefirst and second degrees of mode-mixing such that said second amount ofSOPMD is substantially fixed while said first amount varies.
 129. Thegenerator of claim 125 wherein said controllers are programmed to varysaid PMD spectrum by changing the first and second degrees ofmode-mixing such that said second amount of SOPMD changes at apredetermined rate while said first amount varies.
 130. The generator ofclaim 125 wherein said controllers induce sufficient polarizationmode-mixing between adjacent pairs of stages such that said secondamount changes from a first SOPMD state to a second SOPMD state withoutvarying said DGD value by moving along a constant DGD contour.
 131. Thegenerator of claim 130 wherein said constant DGD contour can bedetermined substantially as follows: τ_(o)=4τ|cos (PM 1)|×|cos (PM 2−PM1)| where τ_(o) is the first amount of DGD, τ is a DGD value of abirefringent stage, PM1 is the first degree of mode-mixing, and PM2 isthe second degree of mode-mixing.
 132. The generator of claim 125wherein said controllers induce sufficient polarization mode-mixingbetween adjacent pairs of stages such that said first amount changesfrom a first DGD state to a second DGD state without varying said secondamount of SOPMD by moving along a constant SOPMD contour.
 133. Thegenerator of claim 132 wherein said constant SOPMD contour can bedetermined substantially as follows: |τ_(w)|=τ^(□.)×{square root}(τ_(1w)²+τ_(2w) ²+τ_(3w) ²), where τ_(1w)=−(1+cos (PM 1))(1+cos (PM 2−PM1)),τ_(2w)=(1+cos (PM 1))(sin 2(PM 2−PM 1)+sin (PM 2−PM 1))−2 sin (PM1), andτ_(3w)=0 and where 51 τ_(w|) is the second amount of SOPMD atsaid optical frequency, τ is a DGD value of a birefringent stage, PM1 isthe first degree of mode-mixing, and PM2 is the second degree ofmode-mixing.
 134. The generator of claim 125 wherein said degrees ofmode-mixing between stages can vary within certain limits such that allconstant SOPMD contours change monotonically.
 135. The generator ofclaim 125 wherein said degrees of mode-mixing between stages can varywithin certain limits such that all said constant DGD value contourschange monotonically.
 136. A method for generating a coherentpolarization mode dispersion (“PMD”) spectrum generator with a PMDgenerator, said generator comprising at least four birefringent stagesin optical series, said stages forming at least three pairs of adjacentstages, and wherein each of said stages comprises a harmonicdifferential group delay (“DGD”) element and a phase-compensatingelement, said method comprising: inducing a degree of polarizationmode-mixing between at least one adjacent pair of said stages.
 137. Themethod of claim 136 wherein said generator further comprises: apolarization mode-mixing element located between at least one pair ofadjacent stages; and a mode-mixing element controller coupled to saidmode-mixing element, and wherein said inducing comprises causing saidcontroller to vary said degree of mode-mixing.
 138. The method of claim137 wherein said at least one pair of adjacent stages comprises all ofsaid pairs of stages, and wherein said inducing comprises varying saiddegrees of mode-mixing between all of said pairs of stages.
 139. Themethod of claim 138 wherein at least one of said mode-mixing elementscomprises an electro-optic element, and wherein said varying said degreeof mode-mixing comprises continuously varying a voltage applied to saidelectro-optic element.
 140. The method of claim 136 wherein each of saidstages has an extraordinary axis, and wherein said inducing comprisesadjusting said extraordinary axes of adjacent stages until an anglebetween said axes is substantially 45 degree to induce maximalpolarization mode-mixing.
 141. The method of claim 136 wherein each ofsaid stages has an extraordinary axis, and wherein said inducingcomprises adjusting said extraordinary axes of adjacent stages until anangle between said axes is either substantially parallel orsubstantially perpendicular to induce substantially zero polarizationmode-mixing.
 142. The method of claim 136 wherein each of said DGDelements is a colorless DGD element, each of said phase-compensatingelements is a locked phase-compensating element, each of said pluralityof optical frequencies is an equally spaced channel of a WDM combspectrum, and wherein said inducing comprises inducing a same amount ofDGD at each of said channels.
 143. The method of claim 136 wherein eachof said DGD elements is a harmonic DGD element, and wherein saidinducing comprises generating differential group delay (“DGD”) andsecond order PMD (“SOPMD”) independently at at least one opticalfrequency by inducing varying degrees of polarization mode-mixingbetween each of said pairs of stages.
 144. The method of claim 143wherein said at least four stages comprises four stages, and said atleast three pairs of stages comprises three pairs of stages, and whereinsaid inducing comprises varying said degrees of mode-mixing between saidstages such that DGD and said SOPMD vary according to a predeterminedtrajectory.
 145. The method of claim 144 wherein said predeterminedtrajectory is selected from a group consisting of a constant DGDtrajectory, a constant SOPMD trajectory, a fixed rate of change of a DGDtrajectory, a fixed rate of change of a SOPMD trajectory, and anycombination thereof.
 146. The method of claim 143 wherein said inducingcomprising varying said DGD and SOPMD within a DGD/SOPMD space in whichsaid DGD and said SOPMD only vary monotonically between zero and arespective maximum.
 147. A method for generating coherent, colorlesspolarization mode dispersion (“PMD”) spectrum with a PMD generatorcomprising at least four birefringent stages in optical series, saidstages forming at least three pairs of adjacent stages, wherein each ofsaid stages comprises a colorless DGD element and a lockedphase-compensating element, said method comprising: inducingpolarization mode-mixing between said stages such that a first amount ofdifferential group delay (“DGD”) and a second amount of second order PMD(“SOPMD”) can be independently generated and controlled at a pluralityof equally spaced optical frequencies in said PMD spectrum.
 148. Themethod of claim 147 wherein inducing comprises controllably andsimultaneously changing said DGD and said PMD.
 149. The method of claim147 wherein said generator further comprises at least one polarizationmode-mixing element between each pair of adjacent stages.
 150. Themethod of claim 149 wherein said at least four stages comprises: (1) afirst stage having an input and an output, (2) a second stage having aninput and an output, wherein said second stage output is opticallycoupled to said first stage input, (3) a third stage having an input andan output, wherein said third stage input is optically coupled to saidsecond stage output, and (4) a fourth stage having an input and anoutput, wherein said fourth stage input is optically coupled to saidthird stage output, wherein said inducing comprises: controlling a firstdegree of mode-mixing between said first and second stages and betweensaid third and fourth stages; and controlling a second degree ofmode-mixing between said second and third stages.
 151. The method ofclaim 150 wherein said controlling said first and second degrees occurssuch that said first amount of DGD is substantially fixed while saidsecond amount of SOPMD varies.
 152. The method of claim 150 wherein saidcontrolling said first and second degrees occurs such that said firstamount of DGD changes at a predetermined rate while said second amountvaries.
 153. The method of claim 150 wherein said controlling said firstand second degrees occurs such that said second amount of SOPMD issubstantially fixed while said first amount varies.
 154. The method ofclaim 150 wherein said controlling said first and second degrees occurssuch that said second amount of SOPMD changes at a predetermined ratewhile said first amount varies.
 155. The method of claim 150 whereinsaid controlling said first and second degrees occurs such that saidsecond amount changes from a first SOPMD state to a second SOPMD statewithout varying said first amount of DGD by moving along a constant DGDcontour.
 156. The method of claim 155 wherein said constant DGD contouris defined substantially according to: τ_(o)=4τ|cos (PM 1)|×|cos (PM2−PM 1)| where τ_(o) is the first amount of DGD, τ is a DGD value of abirefringent stage, PM1 is the first degree of mode-mixing, and PM2 isthe second degree of mode-mixing.
 157. The generator of claim 150wherein said including comprises using said controllers to inducesufficient polarization mode-mixing between adjacent pairs of stagessuch that said first amount changes from a first DGD state to a secondDGD state without varying said second amount of SOPMD by moving along aconstant SOPMD contour.
 158. The generator of claim 157 wherein saidconstant SOPMD contour is defined substantially according to:|τ_(w)|=τ^(□.)×{square root}(τ_(1w) ²+τ_(2w) ²+τ_(3w) ²), whereτ_(1w)=−(1+cos (PM 1))(1+cos (PM 2−PM 1)),τ_(2w)=(1+cos (PM 1))(sin 2(PM2−PM 1)+sin (PM 2−PM 1))−2 sin (PM 1), andτ_(3w)=0 and where |τ_(w)| isthe second amount of SOPMD at said optical frequency, τ is a DGD valueof a birefringent stage, PM1 is the first degree of mode-mixing, and PM2is the second degree of mode-mixing.
 159. The generator of claim 149wherein said inducing comprises varying said degrees of mode-mixingbetween stages within certain limits such that all constant SOPMDcontours change monotonically.
 160. The generator of claim 150 whereinsaid inducing comprises varying said degrees of mode-mixing betweenstages within certain limits such that all said constant DGD valuecontours change monotonically.